Lemma 9.2 [[1]} modified Lemma 3]
Suppose \(x_{k,t}\)  is the interpolation of the discretized process
(REF ). Let \(p_{k,t}\) , \(p_{kt}\)  and \(p_{kt\zeta }\)  denote
its distribution, the joint distribution of \(x_{k,t}\)  and \(x_{k}\) 
and the joint distribution of \(x_{k,t}\) , \(x_{k}\)  and \(\zeta \)  respectively.
Here \(g(x_{k},\zeta )\)  is an estimate of \(\nabla U(x_{k})\)  with noise
\(\zeta \)  such that \(E_{\zeta }g(x_{k},\zeta )=\nabla U(x_{k})\) . Then
\({\displaystyle \frac{d}{dt}H\left(p_{k,t}|\pi _{\mu }\right)\le -\frac{3}{4}I\left(p_{k,t}|\pi _{\mu }\right)+\mathbb {E}_{p_{kt\zeta }}\left[\left\Vert \nabla U(x_{k,t})-g(x_{k},\zeta )\right\Vert ^{2}\right]}.\) 

